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We introduce a non-linear injective transformation τ from the set of non-vanishing normalized Hausdorff moment sequences to the set of normalized Stieltjes moment sequences by the formula$T$[($a$_{$n$})_{$n$=1}^{∞}]_{$n$}= 1/$a$_{1}...$a$_{$n$}. Special cases of this transformation have appeared in various papers on exponential functionals of Lévy processes, partly motivated by mathematical finance. We give several examples of moment sequences arising from the transformation and provide the corresponding measures, some of which are related to$q$-series.

Laser scanning is an effective tool for acquiring geometric attributes of trees and vegetation, which lays a solid foundation for 3-dimensional tree modelling. Existing studies on tree modelling from laser scanning data are vast. However, some works cannot guarantee sufficient modelling accuracy, while some other works are mainly rule-based and therefore highly depend on user inputs. In this paper, we propose a novel method to accurately and automatically reconstruct detailed 3D tree models from laser scans. We first extract an initial tree skeleton from the input point cloud by establishing a minimum spanning tree using the Dijkstra shortest-path algorithm. Then, the initial tree skeleton is pruned by iteratively removing redundant components. After that, an optimization-based approach is performed to fit a sequence of cylinders to approximate the geometry of the tree branches. Experiments on various types of trees from different data sources demonstrate the effectiveness and robustness of our method. The overall fitting error (i.e., the distance between the input points and the output model) is less than 10 cm. The reconstructed tree models can be further applied in the precise estimation of tree attributes, urban landscape visualization, etc. The source code of this work is freely available at https://github.com/tudelft3d/adtree

The irrationality exponent of an irrational number $\xi$, which measures the approximation rate of $\xi$ by rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of $\xi$. Results obtained so far are rather fragmentary and often treated case by case. In this work, we shall unify all the known results on the subject by showing that the irrationality exponents of large classes of automatic numbers and Mahler numbers (which are transcendental) are exactly equal to $2$. Our classes contain the Thue--Morse--Mahler numbers, the sum of the reciprocals of the Fermat numbers, the regular paperfolding numbers, which have been previously considered respectively by Bugeaud, Coons, and Guo, Wu and Wen, but also new classes such as the Stern numbers and so on. Among other ingredients, our proofs use results on Hankel determinants obtained recently by Han.

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